論文の公開元へGeneralized Cousin-I condition and intermediate pseudoconvexity in a Stein manifold
概要
According to the well-known theorem of Cartan-Behnke-Stein [4, 6], every Cousin-I open subset of
C2 is Stein. Here, an open set D in an n-dimensional Stein manifold X is said to be Cousin-I if
any additive Cousin problem has a solution. This condition is equivalent to the injectivity of the
canonical map H 1 (D, O) → H 1 (D, M), where M denotes the sheaf of all germs of meromorphic
functions on D (see Grauert–Remmert [11, p. 137]).
On the other hand, there is an intermediate geometric notion which generalizes pseudoconvexity.
An open set D in an n-dimensional complex manifold X is said to be pseudoconvex of order n − q,
where 1 ≤ q ≤ n, if its complement X \ D has the same continuity as an analytic set of pure
dimension n − q.
The object of this paper is to generalize Cousin-I condition and describe its relation to pseudoconvexity of order n − q. Precisely, we prove that an open set D in an n-dimensional Stein manifold
X is pseudoconvex of order 1 if the canonical map H n−1 (D, O) → H n−1 (D, M) is injective (Theorem 5.1). In the case where n = 2, this result is nothing but the theorem of Cartan-Behnke-Stein
for an open set D in a Stein manifold X of dimension two (see Kajiwara–Kazama [13, Corollary 3]
and Berg [5, Corollary]). Moreover we introduce a new proof of theorem of Eastwood–Vigna Suria. ...
